A conjecture concerning transitive subalgebras of Lie algebras

Abstract

This is to announce the settling of the following conjecture: Given a Lie pseudogroup [ l ] acting transitely on a manifold, is there a finite-dimensional subgroup which also acts transitively? The answer is, in general, no. We give here an example and, in addition, give the Jordan-Holder decomposition of a large class of counterexamples. Finally, we show how these counterexamples occur among general transitive pseudogroups. Following [l] and [2], we work in the category of transitive (filtered) Lie algebras. Details will appear in a forthcoming paper [3]. A transitive algebra L is called minimal if, given a transitive subspace T [ l ] , L is the smallest transitive subalgebra generated by T.